5 In a science fiction story a new type of spaceship travels to the moon. The journey takes place along a straight line. The spaceship starts from rest on the earth and arrives at the moon's surface with zero speed. Its speed, \(v\) kilometres per hour at time \(t\) hours after it has started, is given by $$v = 37500 \left( 4 t - t ^ { 2 } \right) .$$

1. Show that the spaceship takes 4 hours to reach the moon.
2. Find an expression for the distance the spaceship has travelled at time \(t\). Hence find the distance to the moon.
3. Find the spaceship's greatest speed during the journey. Section B (36 marks)